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The curl of a plane vector field has been already defined in section 5.
If the vector field
represent the velocity field of a fluid, the integral in 8.11
defines the flow of the vector field along the curve
.
Definition 8.14
Let
be a smooth curve given by the parametrization
,
where
.
We denote by
a continuous velocity field.
The
flow of
along
is equal to
This integral is called
a flow integral.
If
is closed loop, i.e.
,
then the flow is called the circulation of
around
.
Remark 8.15
This definition is valid for a vector field and a curve in the plane.
Example 8.16 (In the plane.)
Take
and let
be the curve with
parametrization
,
where
.
Figure 4:
A loop and a tangent vector.

 Evaluate
on
:
 Tangent:
 Scalar product:
 Integrate:
Example 8.17 (In the space.)
Take
and let
be the curve with parametrization
,
where
(a loop of an helix, as displayed on Fig.
1).
 Evaluate
on
:
 Tangent:
 Scalar product:
 Integrate:
Definition 8.18
Let
be a smooth curve in the plane. At every point of
is defined a unit normal vector
pointing outwards. Let now
be a vector field in the plane defined over a domain containing the curve
.
The flux of
across the curve
is the line integral
Example 8.19
Take
and let
be the curve with
parametrization
,
where
.
On the curve
:
Therefore:
And we have:
Next: Green's theorem in the
Up: Vector Fields, Work, Circulation
Previous: Work of a force.
Noah DanaPicard
20010530